Idea

A localization , in this sense, of an (∞,1)-categoryCC is a functor L:C→C0L : C \to C_0 to an (∞,1)(\infty,1)-subcategory C0↪CC_0 \hookrightarrow C such that with cc any object there is a morphism connecting it to its localization

c→L(c)
c \to L(c)

in a suitable way. This “suitable way” just says that ff is left adjoint to the fully faithful inclusion functor.

Since localizations are entirely determined by which morphisms in CC are sent to equivalences in C0C_0, they can be thought of as sending CC to the result of “inverting” all these morphisms, a process familiar from forming the homotopy category of a homotopical category.